How would we solve this trigonometric equation using vectors?

Question

$$4\sin\theta +3\cos\theta = 5$$

How would we solve this trigonometric equation using vectors? Since I'm not advanced, I do not truly know where to use vector product.

Regards!

Answer 1

Your equation is equivalent to $$\langle (3,4), (\cos\theta, \sin\theta)\rangle = 5$$

Cauchy-Schwarz inequality gives $$|\langle (3,4), (\cos\theta, \sin\theta)\rangle| \le \|(3,4)\|\|(\cos\theta, \sin\theta)\| = 5$$

so $(3,4)$ and $(\cos\theta, \sin\theta)$ are collinear. Therefore

$$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac43$$ so $\theta= \arctan\frac43 + 2k\pi$ for $k \in \mathbb{Z}$.